What is a negative voltage?
All voltages are measured in a positive or negative direction depending on the 'common' point from which the voltage is referenced.
Voltages are given a positive or negative designation from an arbitrary circuit point normally called 'ground' or 'common'.

Using a battery as a voltage source, if the negative terminal of the battery is connected to 'common' then the voltages in the circuit
will all measure positive from that point of reference. If the positive terminal of the battery is connected to 'common' then the voltages
in that circuit will all measure negative with reference to the 'common' point.

To an engineer, setting this point of 'commanality' in a circuit sets a point of reference for all other voltage measurements.
This simplifies the description of electron flow in a circuit.

Creating a Negative Voltage
For example,
If you take two 1.5V DC cells and place them in series with each other the total voltage is 3V.
If you go between the centre point and the positive rail it will read +1.5V, now move the positve
connection of your multimeter to the negative side and it will read -1.5V.

But for practical purposes, it is not always feasible to add extra power sources and/or batteries to create a negative voltage
for your application, and so there is a much better option.

The MAX232 IC
Typically, the very affordable and commonly available MAX232 IC is used for
RS232 serial communication protocols.
But what you probably didnt know is that
this IC also contains something called a charge pump, an electronic circuit that uses capacitors as energy
storage elements to convert DC voltages into other DC voltages. All you do is give this IC a 5V power supply,
and out comes a negative voltage. This is great if you are already using a MAX232 IC for your serial communication -
and so requires zero new circuitry!

On the IC, use the 'negative output supply voltage Vs-' pin (pin 6 shown below) for up to negative 15V and up to 10mA of current.
10mA isnt enough? Stack multiple MAX232 IC's in parallel for increased current capabilities.

MAX232 PinOut

Negative Voltage Regulators
A voltage regulator keeps a voltage at a defined stable value. Just like a voltage regulator for positive
voltages, a negative voltage regulator can hold a negative voltage.
Other information on voltage regulation available.

Other Methods to get Negative Voltages
Check out this forum post on other methods to generate negative voltages
for your circuit: